Stability analysis of an overdetermined fourth order boundary value problem via an integral identity

TL;DR

This paper investigates the stability of solutions to an overdetermined fourth order boundary value problem, establishing a quantitative estimate of how domain shape deviates from a ball when boundary conditions are slightly perturbed.

Contribution

It introduces an integral identity and a nonlinear weighted trace inequality to derive a stability estimate for near-constant boundary values in a fourth order problem.

Findings

Derived an integral identity for the fourth order Dirichlet problem.

Established a nonlinear weighted trace inequality.

Provided a quantitative stability estimate measuring domain deviation.

Abstract

We consider an overdetermined fourth order boundary value problem in which the boundary value of the Laplacian of the solution is prescribed, in addition to the homogeneous Dirichlet boundary condition. It is known that, in the case where the prescribed boundary value is a constant, this overdetermined problem has a solution if and only if the domain under consideration is a ball. In this paper, we study the shape of a domain admitting a solution to the overdetermined problem when the prescribed boundary value is slightly perturbed from a constant. We derive an integral identity for the fourth order Dirichlet problem and a nonlinear weighted trace inequality, and the combination of them results in a quantitative stability estimate which measures the deviation of a domain from a ball in terms of the perturbation of the boundary value.